Tag Archives: Required Reading

We discussed some key terms that we observed in the first part of the Nova documentary (specifically: fixed point, attractor vs repeller, and strange attractor) and we modeled some of these terms in action with some magnet pendulums. At the close of the period, we also looked briefly at a Solar System simulator found here.

For tomorrow, please read the three sections mentioned on yesterday’s post from Fractals: The Patterns of Chaos

We started watching a Nova documentary on The Strange New Science of Chaos. It’s from 1989, but it has held up well and serves as an excellent introduction to this strange new world of constrained randomness and sensitivity to initial conditions.

For Friday, there are three sections from Fractals: The Patterns of Chaos that I would like you to read:

We started out today announcing the winners for the Fall 2019 Fractal Art Show (congratulations to those winners!)

We spent most of the period playing with F&C Alum Istvan Burbank’s Chaos Game, trying to find specific designs and sharing interesting outcomes. At the end of the period, we also took a brief look at Robert Devaney’s Chaos Apps, in particular his version of the Chaos Game which actually turns the iterative process into a playable game. Keep in mind that this game requires Java, which means it won’t work on your Chromebook. Please let me know if you need some help getting it working!

For Monday, read the Discover Magazine article Chaos Hits Wall Street. While this is an old article (1993!) it addresses a lot of the topics we’ll be discussing in the second part of the course. We’ll see some updated takes on the theories presented in the next articles.

Today, used the Box Count method to find again the dimension of Great Britain (report your findings here) then completed one last project to find calculate the dimension of one of the spiral fractal seen on the last dimension calculation sheet (this took most of the remainder of the period).

We continued yesterday’s applications of the Richardson Plot to the Koch Curve and finally to the coastline of Great Britain, largely confirming Richardson’s findings as included in Mandelbrot’s article. The results of these can be found here.

Also, at the end of class today, we discussed the border between Spain and Portugal and looked at three maps. Take the data below and answer the following questions:

What is the dimension of the border between the two countries?

One country has historically given the length of the border as 987 km, while the other has given a length of 1214 km. Which country is which, and why might this difference have a logical basis (in other words, why might the countries have truly measured the borders in this way? The answer isn’t political!)

Step Size

S

C

Distance measured

100 km

1

7.3

730 km

50 km

2

16.2

810 km

25 km

4

35.4

885 km

10 km

10

93.2

932 km

5 km

20

200.6

1003 km

Remember: for Friday please read pages 61-73 of your new book Fractals: The Patterns of Chaos

For tomorrow, please read Mandelbrot’s revolutionary paper that sparked the recognition of fractals and fractal geometry How Long is the Coast of Britain? You might also want to read this version, where Mandelbrot himself explains how he originally wrote this paper as a “Trojan Horse” to introduce his vision of fractal dimension into the scientific community conversation.

Please sign up for snacks for this week’s Fractal Art Show on Thursday, October 17.

In class today, we worked on our fourth and final dimension classwork sheet (while a handful of stragglers turned in their Fractal Art Show designs) and looked at how the analysis we do to find the dimension of the shapes can be used to recreate them in FractaSketch. Neat!

We also have an important reading assignment: By Friday, October 18, please read Mandelbrot’s revolutionary paper that sparked the recognition of fractals and fractal geometry How Long is the Coast of Britain? You might also want to read this version, where Mandelbrot himself explains how he originally wrote this paper as a “Trojan Horse” to introduce his vision of fractal dimension into the scientific community conversation.

I would also like you to read Pollack’s Fractals, an article from Discover Magazine about the math underlying Jackson Pollack’s famous paintings. For some interesting follow-up reading, check out this article from the New York Times about the use of fractal analysis to examine the authenticity of supposed Pollack paintings and this article from the Science Daily blog suggesting that such an analysis is not scientifically valid.

We started class today with a quick visit to H Courtyard to observe and analyze the dimension of the fractals found there, then came back to class and practiced our Generalized Hausdorf Dimension formula on a new batch of fractals. We’ll spend tomorrow working on our FractaSketch designs (remember that they are due on Friday!)

We spent the day working on FractaSketch. Your submissions for the Fractal Art Show are due next Friday, October 11. Remember, I expect from each of you one entry in three of the following categories:

Fern

Tree (or shrubs, bushes, weeds, etc.)

Spiral

Realistic (other natural phenomena)

Artistic (patterns, designs, etc.)

Again, each student will be submitting three entries, each falling in a separate category. You are welcome to submit more designs if you would like, but they will be placed in a separate “Additional Works” category.

In addition, I will want you to submit the template for one of those official submissions for a template/design matching challenge.

Please feel free to work on your designs outside of class and transfer them to the laptops we’ve been using in class.

We continued to practice finding dimension for a variety of new fractal designs, finishing the first sheet and observing that there is still a bit of a problem with stems with this new definition of dimension. Tomorrow will be a FractaSketch lab day.

We started class with a writing/discussion prompt based on Haldane’s On Being the Right Size, a response to the meme “Would you rather fight one horse-sized duck or 100 duck-sized horses?” The horse-sized duck won the argument, as an application of the central takeaway from Haldane’s article is that artificially scaling a duck up to the size of a horse would increase its weight exponentially faster than its strength could accommodate, resulting in a duck incapable of walking around, let alone flight. We followed this up with a relevant Kurzgesagt (In a Nutshell) video: What Happens if we Throw and Elephant From a Skyscraper (as promised, the follow up — How to Make an Elephant Explode — can be found here).

In the second half of the period, we continued our discussion of the dimension of the various fractals found on our first dimension practice sheet. We found that one of the templates was a cleverly disguised version of the Sierpinski Carpet, and that another–Sierpinski’s Pyramid–has the interesting property of being simultaneously defined as having 1, 2, or 3 dimensions, depending on what type of dimension you use.

Monday, we go back to the computer lab to work on our fractal designs. By Tuesday, read Size and Shape, paleontologist and evolutionary biologist Stephen Jay Gould‘s follow up to Haldane’s essay. If Gould’s name sounds familiar, it’s because he co-developed the idea of (and coined the term for) punctuated equilibrium, a theory of evolution.

We discussed Ivars Peterson’s Ants in Labyrinths at the start of class, noting some interesting passages and talking about questions we had. In particular, I made a note to remember the part towards the beginning, where Peterson suggests an interesting problem with measuring a particular coastline:

Finer and finer scales reveal more and more detail and lead to longer and longer coastline lengths. On a world globe, the eastern coast of the United States looks like a fairly smooth line that stretches somewhere between 2000 and 3000 miles. The same coast on an atlas page showing only the United States […] seems more like 4000 or 5000 miles. […] A person walking along the coastline, staying within a step of the water’s edge, would have to scramble more than 15,000 miles to complete the trip.

This is an important idea. Remember it! We’ll be revisiting it later in the course.

The rest of our time in class was spent working on our fractal designs in FractaSketch. Don’t forget the expectations for the soon-to-be-announced Fractal Art Show! I expect from each of you one entry in three of the following categories:

Fern

Tree (or shrubs, bushes, weeds, etc.)

Spiral

Realistic (other natural phenomena)

Artistic (patterns, designs, etc.)

Again, each student will be submitting three entries, each falling in a separate category. You are welcome to submit more designs if you would like, but they will be placed in a separate “Additional Works” category.

Please feel free to work on your designs outside of class and transfer them to the laptops we’ve been using in class.

Homework: Read On Being the Right Size, an essay written by biologist JBS Haldane in 1926. We will discuss this reading tomorrow. As you read, ask yourself this classic question from the Internet: “Which would you rather fight: one horse-sized duck or 100 duck-sized horses?”

We spent today practicing finding the Hausdorff Dimension for a variety of fractal and template designs, running into some interesting potential problems as well as a few surprising results.

By tomorrow (Thursday), please read the passage Ants in Labyrinths, from Ivars Peterson’s The Mathematical Tourist. As usual, make a note of questions you have and passages you think are significant.

And don’t forget the expectations for the soon-to-be-announced Fractal Art Show! I expect from each of you one entry in three of the following categories:

Fern

Tree (or shrubs, bushes, weeds, etc.)

Spiral

Realistic (other natural phenomena)

Artistic (patterns, designs, etc.)

Again, each student will be submitting three entries, each falling in a separate category. You are welcome to submit more designs if you would like, but they will be placed in a separate “Additional Works” category.

Please feel free to work on your designs outside of class and transfer them to the laptops we’ve been using in class.

We practiced finding dimension with our new definition for a few additional fractals, including seeing a surprising result with the Dragon Curve, then spent the remaining time working on our fractal designs in lab.

For class on Thursday, please read the passage Ants in Labyrinths, from Ivars Peterson’s The Mathematical Tourist. As usual, make a note of questions you have and passages you think are significant.

We looked briefly at Pascal’s Triangle today, and some of the neat patterns that can be found there. I hinted at some hidden fractals that could be found by removing numbers from the triangle, so your homework is to fill in circles in this smaller version that would represent removing every even number from the triangle (remember, we observed that two filled in circles create a filled in one, two empty circles create a filled in one, and an empty and filled circle create an empty one).

We finished our discussion of the ideas inspired by the Jurassic Park excerpt, including looking at a few theories of using fractals to predict financial markets (see the silver and bitcoin articles here if you’d like to read them more closely).

From there, we discussed last night’s assigned reading and used it to form some properties about fractals. In particular:

They demonstrate self-symmetry or self-similarity (each part could be viewed as a scaled-down version of the whole thing)

They are Non-Euclidean (for a Euclidean curve, no matter how wiggly it is, zooming in far enough eventually makes it look linear, but for a fractal zooming in just reveals the same level of detail).

They have fractional dimension (unlike one-dimensional lines, two-dimensional squares, or three-dimensional cubes, fractals live in a space between and could have a non-integer dimension)

This last idea is, of course, pretty wild, and if you feel skeptical about it, you should. Hold on to that skepticism! Let me convince you.

We finished the day making a brief list of ideas of fractals, including snowflakes, ferns, feathers, trees, and river deltas. Your homework is twofold:

Continue to think of examples of fractals in the world around you, and

Read over the excerpt from Edward O Wilson’s book The Diversity of Life: Living Labyrinths. As before, make a note of 2-3 passages that seems significant or questions you have.

We started out today with a discussion about course expectations. This is a pass/fail course but I still expect you to take it seriously. I will often ask you to read an article or do some math work at home, and I expect that work will be done the next day and ready to be discussed. I expect everyone to actively participate in class discussions and engage in the work we do during class time. This is not a study hall, so please don’t bring other work to do during this class or I will ask you to leave.

For your first reading assignment, I’m asking you to read the article distributed in class: Fractals: Magical Fun or Revolutionary Science, from the March 21, 1987 issue of Science News. Take some notes as you read, jotting down the 2 or 3 major points of the article. Pay particular attention to how this article defines the term “fractal”.